I'm working through Rotman's Galois Theory book, and one of the exercises is:
Let $F$ be a field and let $p(x)\in F[x]$ be irreducible. If $g(x)\in F[x]$ is not constant, then either $(p(x), g(x)) = 1$ or $p(x) | g(x)$.
This statement is straightforward to prove, but as far as I can tell, the requirement that $g$ be non-constant is unnecessary. If $g$ is constant, then $(p,g)=1$, so the statement seems to still hold. Or am I missing something?